Chapter 2

(a) Find the sine series of the function \(f(x)=x(\pi-x)\) defined on \([0, \pi]\). (b) Prove that $$ \sum_{n=1}^{\infty} \frac{1}{n^{6}}=\frac{\pi^{6}}{945} $$ (c) Prove that $$ \frac{\pi^{3}}{32}=\frac{1}{1^{3}}-\frac{1}{3^{3}}+\frac{1}{5^{3}}-\frac{1}{7^{3}}+\ldots $$

Set \(f(x)=|\sin x|\) and let $$ f(x) \sim \frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n} \cos n x+b_{n} \sin n x\right] $$ denote the Fourier series of \(f\) on \([-\pi, \pi]\). (a) Calculate the \(a_{n}\) and \(b_{n}\). (b) Set \(g(x)=\sum_{n=1}^{\infty}\left[-n a_{n} \sin n x+n b_{n} \cos n x\right]\). Determine \(g\) and sketch the graph of \(g\) on \([-\pi, \pi]\). (c) Calculate the sums $$ \sum_{n=1}^{\infty} \frac{1}{4 n^{2}-1}, \quad \sum_{n=1}^{\infty} \frac{(-1)^{n}}{4 n^{2}-1}, \quad \sum_{n=1}^{\infty} \frac{1}{\left(4 n^{2}-1\right)^{2}}, \quad \sum_{n=1}^{\infty} \frac{n^{2}}{\left(4 n^{2}-1\right)^{2}} . $$

Let \(f\) be a \(2 \pi\)-periodic piecewise continuous function and let $$ f(x) \sim \frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n} \cos n x+b_{n} \sin n x\right] $$ denote its Fourier series. (a) \(\operatorname{Set} g(x)=f(x+\pi)\) for all \(x \in \mathbb{R}\), and let $$ g(x) \sim \frac{A_{0}}{2}+\sum_{n=1}^{\infty}\left[A_{n} \cos n x+B_{n} \sin n x\right] $$ denote the Fourier series of \(g\). Express \(A_{n}\) and \(B_{n}\) in terms of \(a_{n}\) and \(b_{n}\). (b) Define \(h(x)=f(x) \cos x\), and let $$ h(x) \sim \frac{\alpha_{0}}{2}+\sum_{n=1}^{\infty}\left[\alpha_{n} \cos n x+\beta_{n} \sin n x\right] $$ denote the Fourier series of \(h\). Express \(\alpha_{n}\) and \(\beta_{n}\) in terms of \(a_{n}\) and \(b_{n}\).

Assume \(f\) satisfies the assumptions of Dirichlet's Theorem. Determine the following limits: (a) \(\lim _{n \rightarrow \infty} \frac{1}{\pi} \int_{-\pi}^{\pi} f(t) \sin n t d t\) (b) \(\lim _{n \rightarrow \infty} \frac{1}{\pi} \int_{-\pi}^{\pi} f(t) \sin \left(n-\frac{1}{2}\right) t d t\) (c) \(\lim _{n \rightarrow \infty} \frac{1}{\pi} \int_{-\pi}^{\pi} \frac{f(t)}{t} \sin \left(n-\frac{1}{2}\right) t d t\)

Let \(f \in E\) and $$ f(x) \sim \sum_{n=-\infty}^{\infty} c_{n} e^{i n x} $$ be the complex Fourier series of \(f\). Determine the complex Fourier series of \(f(\bar{x}), \overline{f(x)}\), and \(f(-x)\).

Find the complex Fourier series of \(f(t)=\frac{1}{1-\frac{1}{2} e^{-i t}}\) on the interval \([-\pi, \pi]\).

Find the Fourier series of $$ f(x)= \begin{cases}x-[x], & x \text { is not an integer, } \\ \frac{1}{2}, & x \text { is an integer. }\end{cases} $$ To what values does the Fourier series converge at the points \(x=5, x=3\), and \(x=1.5\) ?

Let \(f \in E\) and assume $$ \frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n} \cos n x+b_{n} \sin n x\right] $$ is the Fourier series of \(f\). Show that if there exist constants \(c\) and \(d\) such that $$ \left|a_{n}\right| \leq \frac{c}{n^{2}}, \quad\left|b_{n}\right| \leq \frac{d}{n^{2}} $$ for all \(n\), then \(f\) may be considered to be continuous on \([-\pi, \pi]\), satisfying \(f(-\pi)=f(\pi)\), and the Fourier series of \(f\) converges uniformly to \(f\) on \([-\pi, \pi]\).

Let \(f, g \in E\) be \(2 \pi\)-periodic functions, and $$ f(x) \sim \sum_{n=-\infty}^{\infty} a_{n} e^{i n x}, \quad g(x) \sim \sum_{n=-\infty}^{\infty} b_{n} e^{i n x} $$ be the complex Fourier series of \(f\) and \(g .\) For each \(x \in \mathbb{R}\) we define $$ h(x)=\frac{1}{2 \pi} \int_{-\pi}^{\pi} f(x-t) g(t) d t . $$ (a) Prove that \(h\) is piecewise continuous and \(2 \pi\)-periodic. (b) Let \(h(x) \sim \sum_{n=-\infty}^{\infty} c_{n} e^{i n x}\) be the complex Fourier series of \(h\). Prove that \(c_{n}=a_{n} b_{n}\) for all \(n \in \mathbb{Z}\).

Let \(f\) be a \(2 \pi\)-periodic piecewise continuous function satisfying $$ \int_{-\pi}^{\pi} f(x) d x=0 . $$ \(\operatorname{Set} g(x)=\int_{0}^{x} f(t) d t\). (a) Prove that \(g\) is \(2 \pi\)-periodic. (b) Let \(\sum_{n=-\infty}^{\infty} c_{n} e^{i n x}\) be the complex Fourier series of the function \(f\) and \(\sum_{n=-\infty}^{\infty} d_{n} e^{i n x}\) the complex Fourier series of the function \(g\). Prove that for all real \(x\) we have the equality $$ g(x)=\sum_{n=-\infty}^{\infty} d_{n} e^{i n x} $$ where \(d_{n}=\frac{c_{n}}{i n}\) for every integer \(n \neq 0\).